Question

The function $ f (x) = |x-10| $ , $ x $ is real number, is

• A. differentiable every where but not continuous at $ x=10 $
• B. continuous everywhere but not differentiable at $ x=10 $
• C. continuous everywhere and differentiable at all points
• D. continuous every where but not differentiable at $ x=0 $

Answer 1

The Correct Option is B

$f(x) = |x - 10| =
\begin{cases}
x-10 & \text{if $x \ge 10$ } \\[2ex]
-(x-10) & \text{if $x < 10$ }
\end{cases}$
Continuity at $x = 10:$
$L.H.S = \lim\limits_{x\to10^{-}} f\left(x\right) = \lim\limits _{h\to0} f\left(10-h\right)$
$ = \lim\limits_{h\to0}-\left(10-h -10\right) = 0$
$R.H.L = \lim\limits_{x\to10^{+}} f\left(x \right) =\lim\limits_{h \to0}f(10+h)$ $=\lim\limits_{h\to0}\left(10+ h -10\right) =0$
Also, $f\left(10\right) = \left|10-10\right| = 0$
Since, $L.H.L. = R.H.L. = f\left(10\right) $
$\therefore f $ is continuous at $ x= 10$
$\Rightarrow f$ is continuous everywhere on real numbers.
Differentiability at $ x = 10$:
$R. H .D. = \lim\limits_{h\to0} \frac{f(10 +h)-f(10)}{h}$
$\lim\limits_{h\to0} \frac{10 + h -10 -0}{h} = 1$
$L.H.D. =\lim\limits _{h\to0} \frac{f(10-h)-f(10)}{-h}$
$=\lim\limits_{h\to0} \frac{-(10-h -10)-0}{-h} =-1$
Since, $L.H.D \ne R.H.D.$
$\therefore f$ is not differentiable at $ x = 10$.